Similarly to the Davison–Meinhardt approach, the semi-orthogonal decomposition for \(W\)  arbitrary follows from the semi-orthogonal decomposition for \(W=0\) . It is important that we formulate a categorical statement as it is not clear how to make the reduction from \(W\)  arbitrary to \(W=0\)  directly in \(K\) -theory. The categories \(\mathbb {M}(d)_w\)  are examples of noncommutative resolutions of singularities of \(X(d)\)  constructed by Špenko–Van den Bergh [1]}. In general, they are not equivalent to \(D^b(\mathcal {Y})\)  for \(\mathcal {Y}\)  a stack.
